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I am trying to check explicitly that the (Compton) amplitude

$$i\mathcal{M} = -ie^2\epsilon^*_\mu(k’)\epsilon_\nu(k)\bar u(p’)\left[\frac{\gamma^\mu \not k\gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k}+\frac{-\gamma^\nu\not{k'}\gamma^\mu+2\gamma^\nu p ^\mu}{-2p\cdot k'}\right]u(p)\tag{5.74}$$

vanishes when we replace either $\epsilon_\nu(k)$ with $k_\nu$ or $\epsilon^*_\mu(k’)$ with $k’_\mu$. This would confirm the Ward identity, as suggested on page 160 of Peskin and Schroeder’s book “Introduction to QFT”.

However, with the replacement $\epsilon_\nu(k) \rightarrow k_\nu$, I obtain:

$$\begin{aligned}k_\nu\mathcal{M}^\nu(k) :&= -e^2\epsilon^*_\mu(k’)k_\nu\bar u(p’)\left[\frac{\gamma^\mu \not k\gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k}+\frac{-\gamma^\nu\not{k'}\gamma^\mu+2\gamma^\nu p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\frac{\gamma^\mu (\not k)^2 + 2\gamma^\mu (p\cdot k)}{2p\cdot k}+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\frac{2\gamma^\mu (p\cdot k)}{2p\cdot k}+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\gamma^\mu+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p)\end{aligned}$$

and the terms in the bracket apparently don’t cancel each other. Are there any further possible simplifications?

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1 Answer 1

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You can further simplify this expression by using the dirac equation $$ 0=(\not p-m)u(p)=\bar u(p')(\not p'-m) $$ and $k+p=k'+p'$. Then the second term can be expressed as $$ 2\not k p^\mu-\not k\not k'\gamma^\mu = 2(\not k'+\not p' - \not p)p^\mu -(\not k'+\not p' - \not p)\not k'\gamma^\mu\simeq 2\not k' p^\mu -(m-\not p)\not k'\gamma^\mu $$ The last equality holds only between the spinors. Then commuting $\not p$ through $\not k' \gamma^\mu$ gives $$ \not p\not k'\gamma^\mu = 2pk'\gamma^\mu-\not k' \not p\gamma^\mu \simeq 2pk'\gamma^\mu - 2\not k' p^\mu + \not k'\gamma^\mu m $$ In total we then have $$ 2\not k p^\mu-\not k\not k'\gamma^\mu \simeq 2pk' \gamma^\mu $$ and the terms in the bracket cancel.

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